On orthogonal polynomials with respect to varying measures on the unit circle
Author:
K. Pan
Journal:
Trans. Amer. Math. Soc. 346 (1994), 331340
MSC:
Primary 42C05
MathSciNet review:
1273539
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Abstract: Let be a system of orthonormal polynomials on the unit circle with respect to and be a system of orthonormal polynomials on the unit circle with respect to the varying measures , where is a sequence of polynomials, , whose zeros lie in The asymptotic behavior of the ratio of the two systems on and outside the unit circle is obtained.
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 [F]
 G. Freud, Orthogonal polynomials, Pergamon Press, New York, 1971.
 [G]
 Ya. L. Geronimus, Polynomials orthogonal on a circle and interval, Pergamon Press, Oxford, 1960. MR 0133642 (24:A3468)
 [LP]
 X. Li and K. Pan, Strong and weak convergence of rational functions orthogonal on the unit circle, submitted..
 [Lo1]
 G. López, Szegö's theorem for polynomials orthogonal with respect to varying measures, Orthogonal Polynomials and their Applications (M. Alfaro et al., eds.), Lecture Notes in Math., vol. 1329, SpringerVerlag, Berlin, 1988, pp. 255260.
 [Lo2]
 , On the asymptotics of the ratio of orthogonal polynomials and the convergence of multipoint Padé approximants, Math. USSRSb. 56 (1987), 207219.
 [Lo3]
 , Asymptotics of polynomials orthogonal with respect to varying measures, Constr. Approx. 5 (1989), 199219. MR 989673 (90h:42036)
 [Lo4]
 , Convergence of Padé approximants of Stieltjes type meromorphic functions and comparative asymptotics for orthogonal polynomials, Math. USSRSb 64 (1989), 207227. MR 954925 (90g:30003)
 [MNT1]
 A. Máté, P. Nevai and V. Totik, Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987), 239282. MR 882423 (88d:42040)
 [MNT2]
 , Extensions of Szegö's theory of orthogonal polynomials, II, Constr. Approx. 3 (1987), 5172. MR 892168 (88m:42044a)
 [P1]
 K. Pan, Strong and weak convergence of orthogonal systems of rational functions on the unit circle, J. Comput. Appl. Math. 46 (1993), 427436. MR 1223008 (94c:42017)
 [P2]
 , On the convergence of rational interpolation approximant of Carathéodory functions, J. Comput. Appl. Math. (to appear). MR 1321080 (95m:65012)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412735399
PII:
S 00029947(1994)12735399
Keywords:
Orthogonal polynomials,
asymptotic properties
Article copyright:
© Copyright 1994
American Mathematical Society
